Legislative Bargaining and Coalition Formation

Journal of Economic Theory 102, 322–353 (2002) doi:10.1006/jeth.2001.2846, available online at http://www.idealibrary.com on

Legislative Bargaining and Coalition Formation 1 Peter Norman Department of Economics, University of Wisconsin at Madison, 1180 Observatory Drive, Madison, Wisconsin 53706 [email protected] Received August 16, 1999; final version received August 20, 2000

The finite horizon version of D. P. Baron and J. Ferejohn’s [1989, Amer. Polit. Sci. Rev. 83, 1181–1206] legislative bargaining model is investigated. With three or more periods, a continuum of divisions is supportable as subgame perfect equilibria. There exist equilibria where coalitions larger than a minimal winning coalition receive strictly positive shares. With sufficiently patient players and a sufficiently long horizon, any interior distribution is supportable as an equilibrium. In contrast, a generic uniqueness result applies when introducing heterogenous time preferences. The unique backwards induction equilibrium in the perturbed game is nonstationary, and neither the original (symmetric) nor the perturbed game provides guidance for equilibrium selection in the infinite game. Journal of Economic Literature Classification Numbers: C73, C78, D72, D78, H49. © 2002 Elsevier Science Key Words: legislative bargaining; coalition formation; generic uniqueness; minimal winning coalitions.

1. INTRODUCTION In an influential paper, Baron and Ferejohn [2] propose a model of legislative voting with endogenous agenda setting. The model has later been generalized and applied to a wide variety of questions in economics and political science. For example, Baron and Ferejohn [3] use the model to analyze the role of committees, Baron [1] studies how legislative equilibria 1 This is a revised version of the first chapter of my dissertation and I stand in great debt to my dissertation supervisor Andrew Postlewaite, Stephen Coate, and Stephen Morris for comments, suggestions, encouragement, and patience. Two referees and an associate editor provided detailed comments and criticism that led to significant improvements of the paper. I also thank George Mailath, Andrea Moro, Motty Perry, Torsten Persson, and seminar participants at the University of Pennsylvania, the University of Wisconsin, IIES Stockholm, IUI Stockholm, and the University of Essex for useful comments. The usual disclaimer applies.

322 0022-0531/02 $35.00 © 2002 Elsevier Science All rights reserved.

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depend on characteristics of the goods provided, Chari et al. [6] use the model in their analysis of split-ticket voting, McKelvey and Riezman [12] use it for an analysis of seniority in legislatures, Merlo [11] studies legislative bargaining in a stochastic environment, and Eraslan [7] uses the model in a study of corporate bankruptcy. The game is a version of Rubinstein’s [15] bargaining model with n \ 3 agents. There are some other differences, but the crucial departure from standard bargaining models is that a proposal is implemented whenever a majority votes in favor of the proposal. Most applications use the version with an infinite time horizon for which a folk theorem applies (Baron and Ferejohn [2]) and get predictive power only by making a particular equilibrium selection, typically the stationary equilibrium. In this paper I study the finite horizon version of Baron and Ferejohn’s model. I show that if there are three or more rounds of bargaining, then a continuum of divisions is supportable as subgame perfect equilibrium outcomes. In fact, any interior division can be supported as a subgame perfect equilibrium outcome if players are sufficiently patient and there are sufficiently many rounds of bargaining. The basic driving force behind these results is that majority rule creates ties. Due to symmetry of the game the ex ante value of the game in the last period is the same for all players. The proposer in the second period from the end needs to bribe only a minimal winning coalition with the discounted value of the game and is therefore indifferent over which players to include in the winning coalition. This in itself implies nonuniqueness, but with three or more rounds of bargaining a more interesting multiplicity arises. If players are selected to be included in the winning majority with different probabilities, then the value of the game in the beginning of the penultimate period is different for different players. Acceptance rules will therefore differ across agents, so some players will be more expensive to bribe than other players. This makes it possible to support a continuum of divisions as equilibrium outcomes even in strategies that are independent of payoff-irrelevant history when mixed strategies are allowed. Players may also pick coalition partners based on history. It is therefore possible to reward agents for rejecting certain proposals by including them in the winning coalition and to punish proposers that deviated by excluding them. This type of strategy may be used to further enlarge the set of equilibrium outcomes to include also divisions with broader support than a minimal winning coalition and, with sufficiently many rounds of bargaining, eventually any interior allocation. The indeterminacy of equilibria is created by ties and if the ties are broken there will be a unique equilibrium. I therefore ask whether a natural perturbation of the payoffs is enough to break the ties. In the paper I allow for heterogeneity in time preferences, but the analysis of the model where

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agents have different probabilities of making proposals is almost identical. Maybe unsurprisingly, I show that when differences in time preferences are allowed there is a unique subgame perfect equilibrium for almost all (vectors of) discount factors. However, whenever the equilibrium is unique each agent has a distinct discounted equilibrium continuation value in every period. Hence, the selection of coalition partners is determinate. Since inclusion in the winning coalition in the next period raises the value of the game in the next period, players that are expensive to bribe in one period are cheap to bribe in the next, so uniqueness implies nonstationarity. One interpretation of the generic uniqueness result is that the symmetric model is a knife-edge case. However, no matter which view one takes on this, the two models in conjunction lead to one unambiguous conclusion. The finite horizon version of the model does not provide any guidance at all for equilibrium selection in the infinite horizon game. This is in sharp contrast with standard Rubinstein-style sequential bargaining where the unique backward-induction equilibrium outcome converges to the stationary equilibrium proposals when the number of rounds in the truncated game goes to infinity. One may think of this as bad news, but the analysis provides some insights as to how sequential bargaining with majority rule (or any other rule different from unanimity rule) differs from the standard setup where unanimous agreement is required. With majority rule, being strong in the future may be a disadvantage since it makes the player more expensive to bribe in earlier stages and therefore more likely to be excluded from the winning coalition.

2. THE SYMMETRIC MODEL The model was initially suggested by Baron and Ferejohn [2] and following their interpretation of the model one may think of a body of legislators, each representing the voters of their district, who are to make a decision about how to distribute some benefits. Time progresses discretely and t=1, ..., T is used to label the periods. I let I={1, ..., n} denote the n | ; ni=1 x i [ 1} denote the set of set of players (legislators) and X={x ¥ R+ potential ‘‘outcomes.’’ Each player i ¥ I is assumed to care only about their own share of the pie and has an instantaneous utility function ui : X Q R given by ui (x)=x i. Payoffs are discounted by the common discount factor d ¥ (0, 1]. The rules of the game are as follows. Proposals. For each t [ T, if the legislature has not already agreed on any proposal one randomly chosen player is recognized to make a proposal

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in the beginning of period t. Irrespective of time and history it is assumed that each player is recognized with probability 1/n. Voting. After a proposal x ¥ X has been made at time t the legislature votes sequentially and according to a predetermined order between accepting and rejecting the proposal (see Fig. 1). If a majority votes to accept, x is implemented and the game ends with payoffs d tx i for each i ¥ I. If a majority decides to reject and t < T the game goes on to the next stage. If a majority decides to reject and t=T the game ends with the zero payoffs being realized for all players. The extensive form is sketched in Fig. 1 for the case with three players, where it should be noted that the voting stage really includes moves by all players. A history at the beginning of time t is given by a list ht = (a1 , a2 , ..., at − 1 ), where ak =(ik , xk , v 1k , ..., v nk ) is the ‘‘action profile’’ for periods k=1, 2, ..., t − 1, ik ¥ I is the identity of the proposer, xk ¥ X is the proposal made, and v jk ¥ {yes, no} is the vote cast by player j in period k. I let Ht denote the set of all possible (beginning of) time t histories. Since voting is sequential, each player knows not only the proposal and ht when voting, but also all previous votes. I let h it denote a history when it is i’s turn to vote in period t which lists the information in ht , the identity of the proposer, the proposal, and all previous votes. The set of all such histories is denoted by H it . A generic pure strategy s i for player i ¥ I is given by s i={p it , v it } Tt=1 , where p it : Ht Q X lists a proposal for each possible history at time t and v it : H it Q {yes, no} describes the voting behavior of player i for every possible history. A generic mixed (behavioral) strategy for i is denoted si and s and s are used in the usual way to denote pure and mixed strategy profiles.

FIG. 1. Sketch of one period of the extensive form.

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3. EXAMPLES To gain intuition and to make the mechanics of the model clear I first consider a few simple examples. All examples assume that there are three players and no discounting. As a benchmark, I begin by constructing a symmetric Markovian equilibrium, which is the finite horizon analogue of the stationary equilibrium in the infinite game usually studied in the literature. I then show that with three rounds of bargaining there is multiplicity in the set of divisions that are supportable as equilibrium outcomes even with history-independent strategies. The result that there is a continuum of distributions supportable as (Markov perfect) equilibrium outcomes follows as a slight generalization of this example. Finally, I show that (again with three periods) there are subgame perfect equilibrium outcomes where all agents get a strictly positive share of the surplus. Hence, the model does not necessarily predict that the cake is divided by a minimal winning coalition. 3.1. The Symmetric Markovian Equilibrium The Last Period. If the proposal in the last period is rejected, the game ends with zero payoffs for everybody. This implies that in any subgame perfect equilibrium the proposer takes everything for herself; that is, player 1 proposes (1, 0, 0), player 2 proposes (0, 1, 0), and player 3 proposes (0, 0, 1), and these proposals are accepted for sure. 2 The Penultimate Period. Since each player gets 1 with probability 13 and 0 with probability 23 if the game continues to the final stage, the continuation value for any player i in the beginning of the last period is 13 . Thus if player j is pivotal in the voting stage of the second period, she must accept if x j > 13 and reject if x j < 13 . Since indifferences must be resolved in favor of acceptance in equilibrium, any proposal where x j \ 13 for at least two agents is accepted and any other proposal is rejected in any subgame perfect equilibrium. Denote by r ij the probability that j is offered 13 by player i when i is recognized to propose. The continuation payoff in the beginning of the second to last period for agent i is then V iT − 1 =13 23+13 r ji 13+13 r ki 13 .

(1)

Hence, if r ij=12 for all i, j (or any configuration of ‘‘inclusion probabilities’’ such that r ij+r kj=1 for all j), the continuation value of the game in the beginning of the second period is 13 for all players. 2 The uniqueness argument combines the fact that pivotal voters must vote in favor of a proposal that benefits them strictly under sequential voting (see Fig. 2) with the usual arguments for why indifferences are resolved in favor of acceptance.

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All Earlier Periods. Consider the game with T \ 3 periods and suppose that after any history of play, each player i randomizes with equal probabilities over which of the remaining players to give 13 to and keeps 23 for herself (that is, player 1 proposes ( 23 , 13 , 0) with probability 12 and ( 23 , 0, 13 ) with probability 12 , and so on). I have already verified that this is consistent with equilibrium at time T − 1. Moreover, the continuation payoff in the beginning of time T − 1 is 13 for each player i. Hence the same argument as in the penultimate period establishes that the voting strategy ‘‘vote to accept if and only if x i \ 13’’ is sequentially rational at time T − 2, implying, that the proposal rule is consistent with equilibrium at time T − 2. The continuation payoff is then again 13 for each player, so it follows by induction that the proposed strategies are consistent with equilibrium. It should be obvious that the playing in accordance with these voting and proposal rules in every period constitutes an equilibrium of the infinite horizon game as well. Indeed, this is what is referred to as the stationary equilibrium of the infinite horizon game, which is the equilibrium selection typically made in applications. 3 3.2. An Asymmetric Markovian Equilibrium I consider the model with three periods, where, again, I for simplicity assume that there are three agents and no discounting. In the second period, let the players follow the proposal rules p 12 (h2 )=( 23 , 13 , 0)

after every h2 ¥ H2 ,

p 22 (h2 )=( 13 , 23 , 0)

after every h2 ¥ H2 ,

p 32 (h2 )=( 13 , 0, 23 )

after every h2 ¥ H2 .

(2)

That is, player 1 always selects player 2, player 2 always selects player 1, and player 3 always select player 1 as her coalition partner. The analysis of the penultimate period of the symmetric equilibrium in Section 3.1 establishes that these proposal rules are consistent with equilibrium.

3 The stationary equilibrium in the infinite horizon game and, (play in all but the last period of) the symmetric Markovian equilibrium in the finite game can, for general (odd) n and an arbitrary d [ 1, be described as follows: In the voting stage, agents vote to accept if and only if they receive at least dn . When making proposals, each agent picks a group of n −2 1 coalition partners by randomizing with equal probabilities over all subsets of n −2 1 players of the remain(n − 1) d ing n − 1. The n −2 1 coalition partners receive dn and the proposer takes 1 − 2n , while all other agents receive nothing.

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Given the proposal rules in (2), the continuation values of the game in the beginning of the second period for players i=1, 2, 3 are V 12 =13 32+13 31+13 13=49 , V 22 =13 32+13 31=39 , and V 32 =13 32=29 .

(3)

The voting rule ‘‘accept if and only if x i \ V i2 ’’ is then sequentially rational for each player. With these acceptance rules player 3 is cheapest, player 2 is the second cheapest, and player 1 is the most expensive player to ‘‘bribe.’’ Hence, the optimal proposals in the first period given the specified continuation play are p 11 =( 79 , 0, 29 ) p 21 =(0, 79 , 29 )

(4)

p 31 =(0, 13 , 23 ), which will be accepted in equilibrium. Hence, I have constructed a subgame perfect equilibrium in Markov strategies that yields equilibrium distributions on the outcome path that is different from the symmetric equilibrium in Section 3.1. Hence, there is a multiplicity of equilibria that goes beyond the identities of the players who are selected for the winning majority. One interesting feature of the asymmetric equilibrium is that the expected payoff for player 1 is lower than for the other two players. At the same time, player 1 is the player who, conditionally on reaching it, has the strongest position in the second round. While this helps player 1 to take a somewhat larger share if making a proposal in the initial period, it is a disadvantage if any of the other players is making a proposal: the other players have less to gain by turning down proposals, so player 1 will simply be excluded from the winning coalition in the first period. That is, being too strong in the future is bad since it makes a player more expensive to bribe. 3.3. Indeterminacy To see that there is indeed a continuum of divisions supportable as sub game perfect equilibrium outcomes, let p be the probability that players behave as in the asymmetric equilibrium constructed above in the second period and (1 − p) be the probability that the players behave as in the

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symmetric equilibrium. With this specification the ordering of the players’ continuation payoffs remains the same for any p ¥ (0, 1] and it is easy to see that by choosing p appropriately any x 1=(s 1, 0, 1 − s 1) with s 1 ¥ [23 , 79] is supportable as an equilibrium outcome (realized if player 1 proposes in the first period). Now let T > 3 and consider the set of equilibria of the T-period game where all players are playing in accordance to the symmetric equilibrium in Section 3.1 from the third period and onward. The (history independent) continuation payoffs in the beginning of period 3 is then 13 for each player, exactly as the unique continuation payoff in the last period of the game. It follows that any division that can be supported in the game with three periods can be supported also in the game with T > 3 periods. It is rather clear that this logic doesn’t rely on either the assumption that there are only three players or the absence of discounting. Indeed, it is easy to show Proposition 1. Suppose that d > 0, T \ 3, and n \ 3. Then there is a continuum of divisions supportable as subgame perfect equilibria of the model. The proof is a direct extension of the construction above, but a sketch is given in Appendix A for completeness. Nothing in the argument hinges on majority rule per se. If q votes are required for acceptance of a proposal the exact same construction can be used to prove that there is a continuum of equilibrium outcomes for any q other than 1 and n. 3.4. Example 2: History Dependent Proposal Rules The purpose of this example is to show that the set of equilibrium outcomes is enlarged even further if history-dependent strategies are allowed. In particular, equilibria where all players receive a positive share of the benefits can be supported by simple punishments and rewards. Again, I consider the case with three agents, three periods, and no discounting. For i=1, 2, 3, I let x i ¥ X be some ‘‘target equilibrium outcome.’’ Without loss I take the first period proposer to be player 1 and assume that, after any history of play, player 1 randomizes with equal probabilities over ( 23 , 13 , 0) and ( 23 , 0, 13 ) if recognized to make a proposal in the second period. Players 2 and 3 on the other hand follow the proposal rules,

˛ ˛

(0, 23 , 13 ) ( 13 , 23 , 0)

if y ] x 1 was proposed in first period if y=x 1 was proposed in first period,

(0, 1 , 2 ) p (h2 )= 1 3 32 ( 3 , 0, 3 )

if y ] x 1 was proposed in first period if y=x 1 was proposed in first period.

p 22 (h2 )= 3 2

(5)

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Since some permutation of ( 23 , 13 , 0) with the proposer getting 23 is proposed after any history of play it follows from the analysis in Section 3.1 that these proposal rules together with the uniquely determined backwardinduction continuation strategies imply Nash equilibrium play in any subgame from the beginning of the second period and on. Now, for a history h2 where x 1 was proposed in the first period and rejected, the continuation payoffs in the beginning of period 2 are V 12 (h2 )=13 32+23 13=188 , V j2 (h2 )=13 32+13 12 13=185

for j=2, 3,

(6)

while if y ] x 1 was proposed in the first period the continuation payoffs are V 12 (h2 )=13 23=29 , V j2 (h2 )=13 23+13 13+13 21 13=187

for j=2, 3.

(7)

Hence, players 2 and 3 will both reject any deviant proposal y in the first period such that y j < 187 , while it is sequentially rational to accept the 5 1j ‘‘target proposal’’ x 1 if x 1j \ 185 . Hence any x 1 such that x 11 \ 11 18 and x \ 18 for one j ] 1 is supportable as an equilibrium outcome when 1 is recognized to make a proposal in the first period. One example of a proposal that can be supported in equilibrium is thus 5 2 ( 11 18 , 18 , 18 ). Hence, three periods are sufficient to generate an equilibrium outcome with broader support than a minimal winning coalition. 4 Another 5 example is ( 11 18 , 18 , 0), which shows that there are inefficient equilibria. The equilibrium in Section 3.1, which is the equilibrium that is typically analyzed, is Pareto efficient, since any proposal where all resources are spent is Pareto efficient. However, extensions of the model have been studied where the players have concave utility functions, either because of risk aversion (see Harrington [9]) or, if the game is interpreted as one where legislators seek funds for local public goods to their districts, due to diminishing returns in spending on local public goods (see Baron [1]). For such preferences equilibria of the form constructed in this example actually have an efficiency rationale since the variability in the share received by any particular agent can be reduced by conditioning play on history as in (5). 5 4 Minimal winning coalitions have appeared as either an essential assumption or the key prediction in most formal models of coalition formation. See for example Riker [14], Shepsle [16], or, more recently, Baron and Ferejohn [2, 3]. For an interesting strategic model where winning majorities are oversized, see Groseclose and Snyder [8]. 5 Norman [13] contains an explicit example where history-dependent strategies are used to construct an equilibrium that Pareto dominates the symmetric equilibrium.

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4. A GENERAL RESULT The ideas in the examples can be used further to increase the set of divisions supportable in equilibrium as the time horizon grows longer. The next proposition provides conditions for when all interior distributions can be supported as a subgame perfect equilibrium. Proposition 2. Suppose that max{2(nn+1 − 1) , `2n/(n − 1)(n − 2)} [ d [ 1 and that there is an odd number of players n \ 5. Then, for any distribution of benefits x such that x i > 0 for all i ¥ I there is a T < . such that for each t \ T there is an equilibrium where player i ¥ I proposes x if recognized to make a proposal in the first period of the game with t periods and x is accepted by a majority. The proof, which is by construction, is given in Appendix B. The idea is to discipline voters by punishing an appropriately chosen group of voters if equilibrium proposals are rejected, to discipline proposers by rewarding an appropriately chosen group of voters for rejecting a deviant proposal, and to make sure that the deviant proposer is excluded in the next period unless he is proposing again. Since it is always possible to make sure that a group of n+1 2 agents gets nothing in the next period unless they are proposing it is rather easy to create incentives for agents to vote to accept proposals even if the share is small. The harder task is to create incentives for agents to propose a division even if their own share is very small. In the proof this is done by ‘‘matching’’ proposals in the sense of giving the n+1 2 agents who get the least under the deviation the shares offered by the deviant proposer (or if this is too little to make it sequentially rational to accept, the lowest share that is consistent with acceptance). Recursive use of these punishment and reward strategies successively erodes ‘‘proposal power,’’ and since the chance of being a proposer in future periods is the source of ‘‘power’’ for the agents who are not currently proposers, this makes it possible to construct an equilibrium where agents are willing to propose and vote to accept proposals in spite of getting an arbitrarily small share. 6 It is important to note that the source of multiplicity is different than in the well-known example of multiplicity in an n-person version of the Rubinstein [15] bargaining model discussed in Herrero [10] and Sutton [17]. Their example relies critically on the infinite horizon and if the game 6 While the language above is similar to repeated-game language it is worth noting that ‘‘to reward’’ means to give tomorrow more than the discounted value of what is offered today. In other words, since the game ends as soon as a proposal is accepted continuation play needs to be more flexible than in typical repeated-game arguments. Hence, the proof resembles the proof of the limit folk theorem in Benoit and Krishna [4] only to the extent that multiplicity in the end is used to successively enlarge the set of continuation payoffs.

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is truncated after t periods uniqueness is restored. The same thing is true for the model in this paper if majority rule is replaced with unanimity, in which case the unique backward induction equilibrium is for the proposer (n − 1) d to give dn to other n − 1 players and keep 1 − n . In contrast, anything strictly in between dictatorship of the proposer and unanimity results in multiple equilibria, so the indeterminacy comes from the voting. The reason that distributions on the boundary cannot always be supported is simple: with positive probability the first-period proposer will propose in every period, so the first-period proposer cannot propose a firstperiod division that gives her less than ( dn ) T > 0 if this proposal would be accepted, since by proposing to take 1 and give nothing to all the others in each period otherwise would be a profitable deviation (this must be accepted in the last period by any subgame perfect strategies). This is in contrast with the infinite horizon game, where all subgames starting with a proposal by some agent look the same. Here proposals on the boundary can be supported by always rejecting anything but the equilibrium proposal with appropriately constructed off-the-equilibrium-path proposal rules. The strategies in the proof were chosen mainly for simplicity and there are several places where more efficient (and more complicated) constructions can be used to ensure faster erosion of proposal power. However, the bound d \ 2(nn+1 − 1) , which is the relevant bound for large n (n \ 9), is tight. In order to defeat the proposal where the proposer takes e and all the others players must prefer the proposal to be rejected. The get n1 −− 1e at least n+1 2 lowest continuation payoff in a group of n+1 2 agents cannot be higher than 2 1−e 2d , so if > there are no feasible continuation strategies that induce n+1 n−1 n+1 incentives to reject the proposal. Hence, d \ 2(nn+1 a necessary condition for − 1) a limit folk theorem to apply. For the restriction on the number of players, the bound is tight and the reason why n=3 does not work is rather simple: if the proposer gives d+e to one player, that player must accept since she could not get more than 1 in the next period.

5. HETEROGENEOUS TIME PREFERENCES The examples and the proof of Proposition 2 exploit that the symmetry of the game makes it possible to punish and reward players in the second to last period without any cost for the proposer. I now ask whether allowing discount factors to vary across players is enough to break all (relevant) ties. The analysis applies with trivial modifications for the model with differences in the probability of making proposals. The case with payoff externalities would presumably be similar as long as externalities are asymmetric. I show that equilibria are now unique for

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almost all choices of discount factors. However, the uniqueness does not help in selecting equilibria. In the most interesting case, when discount factors are close and substantially different from zero, the unique equilibrium will be highly nonstationary. The identities of the (unique) set of agents included in the winning coalition will then depend crucially on how many periods are left in the game. 5.1. Generic Uniqueness Consider the case with three players with d1 > d2 > d3 . Then if player 1 or 2 is to propose in the second stage from the end they will propose to d give 33 to player 3 since player 3 will be the cheapest player to bribe. For the same reason player 3 would select player 2. Now the unique equilibrium continuation values in the beginning of the penultimate period are d d d d d V 1T − 1 =13 (1 − 33 ), V 2T − 1 =13 (1 − 33 )+13 32 , and V 3T − 1 =13 (1 − 32 )+23 33 . So the question of whether the proposals at the third stage from the end are unique or not is a question of whether d1 V 1T − 1 ] d2 V 2T − 1 ] d3 V 3T − 1 or not. Clearly, most choices of (d1 , d2 , d3 ) lead to distinct discounted continuation values. It is intuitive that the discounted continuation values are distinct for most choices of discount factors also with four or more periods and there seems to be no reason to suspect that n players should be any different from 3 players. The next result establishes that this is indeed the case. Proposition 3. For any finite T there exists a set D T … [0, 1] n with full Lebesgue measure such that if (d1 , ..., dn ) ¥ D T then there is a unique equilibrium proposal in each period and for every i ¥ I which will be accepted for sure in any subgame perfect equilibrium. The proof is given in Appendix C, but an informal sketch is instructive. As should be clear from Fig. 2, if continuation values in the beginning of the next period are unique, a proposal is accepted if and only if a majority gets more than their respective continuation values (ties must be broken in favor of acceptance for the usual reasons). 7 Hence, if the set of the ‘‘n −2 1 remaining agents with the lowest discounted continuation payoffs’’ is uniquely defined, then the unique equilibrium proposal when i is recognized is to give these n −2 1 agents their discounted continuation payoffs. A sufficient condition for uniqueness is thus that in each period t \ 2 and for each player i there is a unique equilibrium continuation payoff V it satisfying d iV it ] d jV jt for all j ] i. If this is true at time t, then the optimal proposal at time t − 1 is to select the n −2 1 cheapest players to bribe, give 7 Figure 2 should also make it clear that there is always a trivial multiplicity of equilibria in terms of voting strategies, which is why Proposition 3 is stated in terms of equilibrium proposals.

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these players d iV it , and keep the rest. Hence, if the players are relabeled so that d 1V 1t < d 2V 2t < · · · < d nV nt , then agents i=1, ..., n −2 1 are always included in the winning coalition no matter who the proposer is, agent n+1 2 is included if i [ n −2 1 is proposing, and agents i > n+1 2 are never included in the winning coalition. Hence, the continuation values at time t − 1 can be written in terms of the continuation values at time t and the parameters of the model as

V it − 1

˛

1 2 1 (n − 1) = 1 1 − C d V 2+ dV n 2n 11 1− C d V 2 n n+1

2 n−1 i i 1 1+d iV it − C d jV jt + d Vt n n j=1 n−1 2

j

j t

j

j t

j=1 n−1 2

i

i t

for i <

n+1 , 2

n+1 for i= , 2 for i >

j=1

(8)

n+1 . 2

1

Now, think of d as a free variable and fix d 2, ..., d n. By repeated use of (8) one may express the continuation value for each agent as V it − 1 =a i+b i 1d 1+b i 2(d 1) 2+ · · · +b in − 1(d 1) T − t − 1.

(9)

Hence, any failure of uniqueness must correspond to some d 1 that solves a polynomial. The relevant polynomial will depend on d 1, but only to the extent that the ranking matters. Since there are finitely many ways to rank a finite set of objects there are finitely many polynomials to consider and each has a finite number of roots. Finally, the coefficients of each polynomial are continuously differentiable in (d 2, ..., d n), so discount factors where uniqueness can fail belong to a finite collection of smooth manifolds of dimension n − 1, a set of measure zero.

FIG. 2. Voting with unique continuation payoffs when x 1 > d 1V 1t , x 2 < d 2V 2t , and x 3 > d 3V 3t .

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5.2. Implications for Equilibrium Selection in the Infinite Game As is well-known and mentioned previously in the paper, majority rule is not needed to get a folk theorem in the infinite horizon game. Even in Rubinstein style bargaining with three or more players anything can be supported as a subgame perfect equilibrium in the infinite horizon game. However, here the (unique) stationary equilibrium may be viewed as a limit of unique backward-induction equilibria of truncated games (see Sutton [17]). Here, the situation is similar, with one important difference. In Rubinstein bargaining the equilibrium proposals converge, while this cannot be true in the model with majority rule if the equilibrium is unique except under exceptional and rather uninteresting circumstances. The idea is much the same as the example in Section 3.2, where the player that got favorable treatment in the second period was at a disadvantage since she was too expensive to bribe in the first period. This logic is reflected in (8) in that the continuation value of the agents with low indices consists of one part which is due to the share taken if the agent is recognized to propose at time t − 1 and in addition to this they get the discounted value of the continuation payoff at time t for sure if any other agent is called to propose at time t − 1. Agents with high indices, on the other hand, only get a positive share if they are called to propose at time t − 1. Unless there are very large (and systematic) differences in time preferences this means that the low index agents will be the agents that are the most expensive to bribe in period t − 2. Hence, the identity of the agents that are included in the winning coalition will change in every period, so the proposals do not converge. The conclusion of this is that the perturbed model is as silent on equilibrium selection for the infinite horizon game as is the symmetric model. Of course, there are alternative criteria for equilibrium selection and other types of arguments have been proposed in favor of the stationary equilibrium, the most appealing perhaps being the simplicity. Indeed, Baron and Kalai [4] show that the stationary equilibrium is the simplest equilibrium of the game in the sense that it requires the smallest number of states in order for an automaton to play it among all equilibrium strategies. 6. CONCLUDING REMARKS For the finite horizon version of the popular model of legislative bargaining due to Baron and Ferejohn [2] I have shown that there is a continuum of divisions supportable as an equilibrium outcome for any (common) discount factor as long as there are three or more rounds of bargaining. Moreover, if the players are sufficiently patient any distribution of benefits where all agents receive a strictly positive share can be supported as an equilibrium outcome if there are sufficiently many rounds of

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bargaining. I also generalized the model to allow different players to have different time preferences and showed that this perturbed model has a unique equilibrium for almost all choices of (vectors of) discount factors. A natural response to the generic uniqueness is to view the symmetric game as special and dismiss the model completely since the results are so wildly different from the more generic perturbed model. However, there are interesting circumstances when this is not so obvious. In particular, a natural line of research is to model representative democracy (as in Chari et al. [6]) by assuming that citizens elect candidates and thereby choose the preferences for the legislators who play a legislative bargaining game after the election. Since the representatives of half the districts are excluded for sure from the winning coalition (unless proposing) if the equilibrium in the legislative game is unique, a reasonable conjecture for such a setup is that preferences of the elected legislators correspond to the ‘‘nongeneric’’ symmetric case in equilibrium.

APPENDIX A: PROOF OF PROPOSITION 1 Proof. Arguing as in Section 3.1 one shows that there are (symmetric) equilibrium strategies such that the continuation value is 1n in the beginning of period 3 for all players after any history of play (see Baron and Ferejohn [2] for details). Given these equilibrium continuation strategies, each agent must in equilibrium bribe n −2 1 others with exactly dn and keep the rest when proposing in period 2. Suppose each player selects the n −2 1 remaining players with the lowest indices with probability p and randomizes with equal probabilities over the remaining agents with probability (1 − p). The value of the game in the beginning of the second period is then

˛

1

2

1

2

1

1 (n − 1) d (n − 1) d 1 d 1− + p +(1 − p) n 2n n n 2 n n+1 for i < , 2 1 (n − 1) d (n − 1) 1 d 1− + n 2n n 2 n V i2 (p)= n+1 for i= , 2 1 1 d (n − 1) d (n − 1) 1− + (1 − p) n 2n n 2 n n+1 for i > . 2

1

2

2

(A1)

LEGISLATIVE BARGAINING

337

Given any d > 0, n \ 3, and any p > 0 we see from (A1) that V (n+1)/2 (p) < 2 n+3 (p) for i ¥ { , ..., n}. Hence V i2 (p) for i ¥ {1, ..., n −2 1} and V i2 (p) < V (n+1)/2 2 2 if agent i ¥ {1, ..., n −2 1} is called to propose in the first period she will propose to keep 1 − d ; ni=(n+3)/2 V j2 (p) and give dV j2 (p) to agents j= n+3 j 2 , ..., n. Depending on the choice of p, V 2 (p) can take on any value (n − 1) d 1 1 between n (1 − 2n ) and n , implying that there is a continuum of equilibrium proposals. The argument goes through with obvious modifications for other proposers. L

APPENDIX B: PROOF OF PROPOSITION 2 For the purposes of the proofs of Propositions 2 and 3, and only for these two proofs, I abuse previously used notation and use t to indicate the period with time counted backward with t=1 being the last period. A generic proposal is denoted x=(x 1, ..., x n) or, if more than one proposal has to be considered at the same time, y=(y 1, ..., y n). Subscripts denote time and superscripts on specific shares of a given proposal are used to indicate who is the receiver of a certain share. When specifying proposal rules, however, an index is needed also for the proposer. I use the notation p it : Ht Q X to denote a generic proposal rule at time t for agent i. To index the particular shares under a specific proposal rule I use two superscripts where the first is the identity of the proposer and the second is the identity of the receiver: i.e., p ijt (ht ) denotes the share received by agent j if i is the proposer at time t after history ht . Lemma B1. Let ct , dt \ 0 and suppose that for each i ¥ I and each J … I 0 {i} such that |J|=n −2 1 , there is an equilibrium in the game with t periods where i proposes xt given by

˛

ct x = dt 0 j t

to j=i to all j ¥ J to all j ¨ J 2 {i}

(B1)

and that xt is accepted by a majority in the first period. Then, for any feasible proposal x such that (1) x i \ dn for one agent i, (2) x j \ dn ct for n −2 1 agents j ] i, there is an equilibrium in the game with t+1 periods in which x is accepted if proposed. Proof. Consider some x satisfying the conditions (l) and (2) and relabel the players so that x 1 \ dn and x j \ dct /n for j=2, ..., n+1 2 . After any history

338

PETER NORMAN

ht when x was proposed and rejected at time t+1, let the proposer i at time t follow the rule

˛

ct

to j=i

n+1 d if i [ let p ijt (ht )= t 2 0

to all j >

n+1 2

to all other agents j [

n+1 , 2

and

˛

cy

n+1 d if i > let p ijt (ht )= t 2 0

to j=i n+1 and j=1 (B2) 2 n+1 . to all agents j=2, ..., 2 to all other j >

Only permutations of (B1) are proposed under (B2), so the proposal rule is consistent with subgame perfect equilibrium play in all subgames from time t by assumption. The continuation payoff in the beginning of period t (before the identity of the proposer is drawn) is then 1 n−1 1 V1 = ct + d [ n 2n t n

for j=1

1 VL = ct n

for j=2, ...,

1 n−1 2 − ct VH = ct + dt [ n n n

n+3 for j= , ..., n, 2

n+1 2

(B3)

where the inequalities follow since dt [ 2(1 − ct )/(n − 1) for (B1) to be feasible. Since the continuation values in the beginning of time t are independent of voting behavior and since x 1 \ dn \ dV1 and x j \ dVL =dct /n the voting rule ‘‘vote to accept independently of voting history’’ is sequentially rational for agents 1, ..., n+1 2 . Hence there are strategies consistent with equilibrium play in every subgame following the proposal x in the beginning of time t+1 such that x is accepted. L

339

LEGISLATIVE BARGAINING

Lemma B2. Suppose that for each i ¥ I and each feasible proposal xt such that x it \ ct x jt \

for agent i

d n

for some agent j (possibly j=i)

x kt \ dt

for at least

(B4)

n−1 agents k ] i 2

there is an equilibrium in the game with t periods where x it is proposed and accepted by a majority if i is recognized in the first period. Then, for any proposal y such that there exists a set J of n+1 agents j ] i such that 2 yj ;j ¥ J max{dt , d } < 1 − ct , there are sequentially rational continuation strategies such that (1) y is rejected if proposed by i at time t+1 and (2) the discounted expected payoff for the proposer i is dct /n. Proof. Without loss, relabel the agents so that agent n is the proposer at time t+1 and y 1 [ y 2 [ · · · [ y n − 1. Given this labeling there exists a group yj agents different from the proposer such that ;j ¥ J max{dt , d } < J of n+1 2 1 − ct if and only if

3

n+1 2

4

yj < 1 − ct . d

C max dt , j=1

(B5)

Suppose that each i ¥ I follows the proposal rule

˛ ˛

n+1 if i [ let p ijt (ht )= 2

y y 1+max 3 d , 4 − C max 3 d , 4 d d n+1 2

i

j

t

t

j=1

3

max dt ,

yj d

to j=i

4

n+1 2 to all other agents, to all other j [

0 ct

4 n+1 if i > let p (h )= 2 y 1 − C max 3 d , 4 − c d ij t

t

3

max dt ,

yj d

n−1 2

j

t

j=1

0

t

(B6)

to j=i to all j [

n−1 2

n+1 to j= 2 to all other agents

340

PETER NORMAN

if recognized to make a proposal at time t. Consider first the case when ij ii i [ n+1 2 is proposing at time t. Clearly p (ht ) \ 0 for all j (p (ht ) > 0 follows ij from (B5)) and ;j p (ht )=1, so the (B6) specifies a feasible proposal. From (B5) it follows that y y p (h )=1+max 3 d , 4 − C max 3 d , 4 > c , d d i

ii

t

n+1 2

t

j

t

t

(B7)

j=1

so the first inequality in (B4) is satisfied. Since the shares sum to unity and 2 d ij only n+1 2 agents get a positive share p (ht ) \ n+1 > n for at least one j ¥ I, so the second inequality in (B4) is also satisfied. Finally, p ij(ht ) \ dt for all n−1 i agents in {1, ..., n+1 2 } 0 {i} (i.e., 2 agents), so p t (ht ) satisfies the three conditions in (B4), and (B6) is consistent with subgame perfect equilibrium play where p i(ht ) is accepted at time t after all histories consistent with ii (B5). Similarly, if i > n+1 2 is recognized in period t, then p t (ht )=ct , immen+3 diately establishing the first condition in (B4). Second, 2 agents split 1 2 and n > 3, so there must be some j such that p ij(ht ) > n+3 \ 1n > dn , so the ij second condition in (B4) holds as well. Finally p t (ht ) \ dt for agents n+1 1, ..., n+1 2 (where (B5) is used to derive the conclusion for agent 2 ), so also in this case all conditions in (B4) are satisfied, implying that p it (ht ) is consistent with subgame perfection after all relevant histories. In both cases, ij j each agent j [ n+1 2 receives a share p (ht ) \ y /d independently of who is the ii j proposer at time t and p (ht ) > y /d, so with probability 1n they receive strictly more. The discounted expected payoff if y is rejected is thus strictly larger than y j for j=1, ..., n+1 2 . Hence, the voting strategy ‘‘always vote to reject y’’ is sequentially rational for j=1, ..., n+1 2 , implying that there is an equilibrium where y is rejected if proposed for any y satisfying (B5). The proposer at time t+1 (n) gets ct if proposing again at time t and 0 otherwise, so her discounted expected payoff from proposal y is dct /n, which completes the proof. L

Lemma B3. Suppose that the assumptions of Lemma B2 are satisfied. Then, for any proposal y such that y j < dn for one agent i and y j < d(2 − ct )/n for n −2 1 agents j ] i there are sequentially rational continuation strategies such that (1) y is rejected if proposed by i at time t+1 and (2) the discounted expected payoff for i is dct /n. Proof. As in Lemma B2 I relabel the agents so that agent n is the proposer at time t+1 and y 1 [ y 2 [ · · · [ y n and index the proposer at time t

341

LEGISLATIVE BARGAINING

as agent i. If y is rejected at time t+1 I assume that the proposer at time t, i, follows the rule

˛ ˛

ct n+1 2(1 − ct ) if i [ she proposes p ijt (ht )= 2 n−1 0 ct n+1 2(1 − ct ) if i > she proposes p ijt (ht )= 2 n−1 0

to herself n+1 2 to all other agents (B8) to herself n+1 to j=2, ..., 2 to all other agents, to all other j [

and that p it (ht ) is always accepted, which by assumption is consistent with subgame perfect equilibrium play from time t on (because (B4) are satisfied). If y is rejected, the continuation payoff in the beginning of period t is 1n for j=1 and (2 − ct )/n for j=2, ..., n+1 2 . Hence it is consistent to vote against y for all with sequential rationality for agent 1, ..., n+1 2 voting histories (since y 1 < dn and y j < d(2 − ct )/n for j=2, ..., n+1 2 ). Again, the proposer at time t+1, n, gets ct if called to propose at time t and 0 otherwise, so the discounted expected payoff for the proposer is dct /n. L Lemma B4. Suppose that the assumptions of Lemma B2 are satisfied. Then, for any proposal y such that y j < d( 1n+((1 − ct )/n)((n − 1)/(n+1))) for n+1 agents j different from the proposer there are sequentially rational 2 continuation strategies such that (1) y is rejected if proposed at time t+1 and (2) the discounted expected payoff for the proposer is dct /n. Proof. Again, relabel the agents so that n is the proposer at t+1 and y 1 [ y 2 [ · · · [ y n − 1. If i [ n+1 2 is recognized in period t the proposal rule is still in accordance with (B8), but if i > n+1 2 proposes at time t she follows the rule

˛

p ijt (ht )=

ct 2(1 − ct ) n−1

0

to the proposer at time t, n−1 to randomly selected agents 2 n+1 from 1, ..., , 2 to all other agents.

3

4

(B9)

342

PETER NORMAN

Given this proposal rule the probability for an agent j [ n+1 2 to be included is called to propose at time t is in the winning majority when agent i > n+1 2 n−1 n+1 , so the continuation value of the game for agents 1, ..., in the n+1 2 beginning of period t is 1 (n − 1) 2(1 − ct ) (n − 1) (n − 1) 2(1 − ct ) VH = ct + + n 2n (n − 1) 2n (n+1) (n − 1) 1 (1 − ct ) (n − 1) . = + n n (n+1)

(B10)

Hence y j < dVH for agents 1, ..., n+1 so it is consistent with sequential 2 to vote to reject y independently of previous rationality for agents 1, ..., n+1 2 votes. The outcome of such voting strategies is that y is rejected and, again, the proposer at time t+1, n, gets ct if called to propose again at time t and nothing otherwise, so the discounted expected payoff is dct /n. L For ease of stating the next result I define the functions

1

2 (n − 3) d 1 1 n − 1 (1 − c) 2 H(c) — 1 − d(1 − c) − + . 2 n n+1 n

G(c, d) — 1 − d 1 − c −

(n − 1) (n − 3) d d − (2 − c), 2 2n

(B11)

Lemma B5. Suppose that the assumptions of Lemma B2 are satisfied and (n − 1) d also suppose that ct [ 1 − 2n and dt [ dn. Then, for any i ¥ I and any feasible proposal x such that

x i \ max

3 dcn , G(c , d ), H(c ) 4 t

t

t

t

xj \

d n

for some j ¥ I( possibly j=i)

xk \

dct n

for at least

(B12)

n−1 agents k ] i 2

there is a subgame perfect equilibrium in the game with t+1 periods in which x is proposed and accepted by a majority if i is recognized in the first period.

343

LEGISLATIVE BARGAINING

Proof. Consider some x satisfying (B12). By assumption, any proposal satisfying (B4) is an equilibrium outcome of the t period game, implying that any perturbation of the proposal (B1) is an equilibrium proposal in the game with t periods. Since Eqs. (B12) imply that x j \ dn for one j and xk \ dct /n for at least n −2 1 agents k ] j, Lemma B1 implies that there are sequentially rational continuation strategies such that x is accepted if proposed at time t+1. It remains to check whether equilibrium strategies can be constructed so that i is willing to propose x if recognized at time t+1. I let y be an arbitrary deviation from x and relabel the agents so that n is the proposer at time t+1 and y 1 [ y 2 [ · · · [ y n − 1. Case 1. First suppose that y is such that any of the conditions

3

n+1 2

C max dt , j=1

4

yj [ 1 − ct d y1 < y

n+1 2

d n


(B13) y

and

n+1 2

<

(2 − ct ) d n

− 1) 2 1 1n+(1 −n c ) (n(n+1) t

(B14) (B15)

is satisfied. Lemma B2 established that there are sequentially rational continuation strategies so that y is rejected if proposed if (B13) is satisfied and Lemma B3 and B4 guarantee that the same thing is true under conditions (B14) and (B15) respectively. In all three cases continuation play can be constructed so that the expected discounted payoff for n (the proposer at time t+1) is dct /n. I assume continuation play after a deviant y satisfying (B13), (B14), or (B15) is such that n receives dct /n and since x \ dct /n it follows that no such deviation is profitable. Case 2. Next consider some y where n+1 2

3

C max dt , j=1

4

yj \ 1 − ct d

and

y

n+1 2

\

Since

d

1

(2 − ct ) 1 (1 − ct ) (n − 1) >d + n n n (n+1)

2

(2 − ct ) d . n

(B16)

344

PETER NORMAN

neither condition (B13), (B14), or (B15) is satisfied. I assume that continuation play after a rejection of y is in accordance with any sequentially rational strategies where agent n gets an expected discounted payoff of dct /n (for example, the strategies in Lemmas B2, B3, and B4 all work). It follows that the deviation must be accepted in order to be profitable, (n − 1) d generating a payoff x n to the proposer at time t+1. Since ct [ 1 − 2n it (n − 1) d (2 − ct ) d (n+1)/2 \ d n > n \ dt . This together follows that 2 − ct \ 1+ 2n > 1, so y with the first inequality in (B16) implies that

3 yd 4 y y n−1 1 = C max 3 d , 4+ [ d+ C y d d 2 d n−1 S C y \ d 11− d − c 2. 2 n+1 2

j

1 − ct [ C max dt , j=1 n−1 2

j

n+1 2

n+1 2

t

j

t

j=1

j=1

n+1 2

j

t

t

(B17)

j=1

Moreover, since y 1 [ y 2 [ · · · [ y n − 1, the second inequality in (B16) (n − 3) d −1 implies that ; nj=(n+3)/2 y j \ 2n (2 − ct ). Hence, a proposal that satisfies (B16) gives the proposer no more than n+1 2

1

n−1

yn [ 1 − C yj − C yj [ 1 − d 1 − j=1

n+3 j= 2

2

n−1 (n − 3) d dt − ct − (2 − ct ) 2 2n

=G(ct , dt ) [ x n.

(B18)

Hence, there is no profitable deviation satisfying (B16). Case 3. The final possibility is that y satisfies n+1 2

3

C max dt , j=1

4

yj \ 1 − ct d y

n+1 2

\d

and

y1 \

d n

(n − 1) 2 1 1n+(1 −n c ) (n+1) . t

and (B19)

Again I let continuation strategies be such that the proposer has no incentive to propose a proposal that is rejected, so that in order for y to be a profitable deviation, y n > x n. By assumption, dt [ dn , so dn [ y 1 [ y j for all

345

LEGISLATIVE BARGAINING yj

j \ 1. Thus, ; (n+1)/2 max{dt , d }=1d ; (n+1)/2 y j. Moreover, the third condij=1 j=1 (n − 3) d 1 (1 − c ) (n − 1) n−1 j tion in (B19) implies that ; j=(n+3)/2 y \ 2 ( n+ n t (n+1) ), so a proposal satisfying (B19) gives the proposer a share no larger than n+1 2

n−1

n

y [ 1 − C y j − C y j [ 1 − d(1 − ct ) − n+3 j= 2

j=1

1

(n − 3) d 1 (1 − ct ) (n − 1) + 2 n n (n+1)

=H(ct ) [ x n.

2

(B20)

A proposal that does not satisfy either (B16) or (B19) necessarily satisfies either (B13), (B14) or (B15), so this concludes the proof. L Lemma B6. If d \ 2(nn+1 − 1) , then H(c) < c for all c ¥ [0, 1]. (n − 1) (1 − c)

Proof. Since 1 − c \ 0 and 1n+(n+1) n \ 0 it follows from direct inspection of the definition of H(c) in (B11) that H(c) is decreasing in d for any c. For any d \ 2(nn+1 − 1) and c ¥ [0, 1] we therefore know that

1 2 2n(n+1)+(n − 1)(n − 3) 2 2(n+1)+(n − 3) 2 =c 1
H(c) [ 1 −

=c

(n − 3)(n+1) 1 n − 1 (1 − c) n+1 (1 − c) − + 2(n − 1) 4(n − 1) n n+1 n

(3n − 1)
(B21)

since n \ 5. L

d n

2n Lemma B7. Suppose that `(n − 1)(n − 2) [ d [ 1, ct [ 1 − (n − 1) d (n − 1) d (1 − 2n ). Then, G(ct , dt ) < 1 − 2n .

(n − 1) d 2n

, and dt [

Proof. G is increasing in both arguments, so

1

G(ct , dt ) [ G 1 −

1

(n − 1) d d (n − 1) d , 1− 2n n 2n

22

1 (n −2n1) d − (n −2 1) dn 1 1 − (n −2n1) d 22 (n − 3) d 1 (n − 1) d 2 − 1+ 2n 2n (n − 1) d 2 2 (n − 3) d d (n − 3)(n − 1) =1 − d 11 − − . 2n 2n 4n =1 − d

2

2

2

(B22)

346

PETER NORMAN 2

(n − 1)

(n − 1)

For d=1, this gives G(ct , dt ) [ 1 − 2n +1n ( 5n −2nn − 2 ) < 1 − 2n , where the second inequality follows since 5n − n 2 − 2 < 0 for n \ 5. Hence, the claim is 2n true for d=1. For the case with `(n − 1)(n − 2) [ d < 1, we observe that d \ 2n `(n − 1)(n − 2) implies that 2n (n − 1) (n − 1) 1 (n −2n1) d 2 \ (n − 1)(n = − 2) 4n 2n(n − 2) 2

2

2

and d 2(n − 1)(n − 3) 2n(n − 1)(n − 3) (n − 3) \ 2 = . 4n 2 4n (n − 1)(n − 2) 2n(n − 2)

(B23)

Combining (B22) and (B23) we find that G(ct , dt ) [ 1 − d =1 −

(n − 1) 2 (n − 3) d (n − 3) 1 2n(n − − − 2) 2n 2n(n − 2)

(n − 1) d (n − 1) d (d − 1) (n − 3) + < 1− , 2n n 2(n − 2) 2n

(B24)

since d < 1. L (n − 1) d

Lemma B8. For any feasible x such that x i \ 1 − 2n for agent i and (n − 1) d j x \ dn (1 − 2n ) for n −2 1 agents j ] i there is an equilibrium in the game with three periods in which i proposes x if recognized in the first period and x is accepted by a majority. Proof. A standard backward-induction argument (see Proposition 1 in Baron and Ferejohn [2]) establishes that in any equilibrium each agent i selects a set J … {1, ..., n} 0 {i} with |J|=n −2 1 of agents to ‘‘bribe’’ and proposes

˛

1−

p kj 2 (h2 )= d n 0

(n − 1) d 2n

for j=i for all j ¥ J

(B25)

for all j ¨ J 2 {i},

if i is called to make a proposal at t=2. This proposal is accepted in any equilibrium. Since the proposer is indifferent between any J, JŒ … {1, ..., n} 0 (n − 1) d {i}, the assumptions of Lemma B1 are satisfied at t=2 for c2 =1 − 2n

347

LEGISLATIVE BARGAINING

and d2 =dn. Consider an arbitrary x satisfying the inequalities in the statement of the lemma. Assume that continuation play is such that x is accepted if proposed, which, by Lemma B1, is consistent with sequential rationality in all subgames following the proposal of x. After any other proposal y ] x assume that each agent randomizes with equal probability (n − 1) d over all other agents to select n −2 1 agents to receive dn and keeps 1 − 2n . Clearly, this is consistent with (B25) and the continuation payoff in the beginning of period 2 is then 1n for all agents after a history when y ] x was proposed in period 3. Hence, if some y ] x is proposed at time 3 the voting rule ‘‘accept y ] x if and only if y j \ d/n’’ is sequentially rational for all (n − 1) d agents j ¥ I, implying that the proposer must give at least 2n to the other (n − 1) d players to get a proposal y ] x accepted. Thus y i [ 1 − 2n [ x i for any proposal that would be accepted and the discounted expected payoff from (n − 1) d a proposal that is rejected in equilibrium is dn < 1 − 2n . Hence, this constitutes an equilibrium where x is proposed and accepted at time t. Since x is an arbitrary allocation satisfying the inequalities in the claim this ends the proof. L 2n Lemma B9. Suppose that d \ max{2(nn+1 and let the − 1) , `(n − 1)(n − 2) } (n − 1) d (n − 1) d . . sequences {ct } t=3 and {dt } t=3 be given by c3 =1 − 2n , d3 =dn (1 − 2n ) dct − 1 d and ct =max{ n , G(ct − 1 , dt − 1 ), H(ct − 1 )}, dt =n ct − 1 for each t \ 4. Then, 0 < ct < ct − 1 and 0 < dt [ dt − 1 for each t > 4.

(n − 1) d

2n Proof. d \ `(n − 1)(n − 2) so, by Lemma B7, G(c3 , d3 ) < 1 − 2n =c3 . dc3 Lemma B6 guarantees that H(c3 ) < c3 since d \ 2(nn+1 − 1) and since 0 < n < c3 dc it follows that 0 < c4 =max{ n3 , G(c3 , d3 ), H(c3 )} < c3 and 0 < d4 =dn c3 = (n − 1) d d n (1 − 2n )=d3 , so the claim is true for t=4. Next, suppose that 0 < ct < ct − 1 and 0 < dt [ dt − 1 for some t > 4. The function G is strictly increasing in both arguments, so G(ct , dt ) < G(ct − 1 , dt − 1 ) and since c G(ct − 1 , dt − 1 ) [ max{d tn− 1 , G(ct − 1 , dt − 1 ), H(ct − 1 )}=ct it also follows that dc G(ct , dt ) < ct . Lemma B6 guarantees that H(ct ) < ct and 0 < nt < ct since dc ct > 0. Taken together this means that 0 < ct+1 =max{ nt , G(ct , dt ), H(ct )} < d ct and since 0 < ct < ct − 1 it also follows that dt+1 =n ct < dn ct − 1 =dt − 1 . The result follows by induction. L

2n Lemma B10. Suppose that d \ max{2(nn+1 − 1) , `(n − 1)(n − 2) } and let the . . sequences {ct } t=3 and {dt } t=3 be defined as in Lemma B9. Then, for each t \ 3, each i ¥ I, and any feasible proposal xt such that (1) x it \ ct for agent i, (2) x jt \ dn for one agent j (possibly j=i), (3) x kt \ dt for at least n −2 1 agents

348

PETER NORMAN

k ] i, there exists an equilibrium in the game with t periods in which xt is proposed and accepted in the first period if agent i is recognized to make a proposal. Proof. Lemma B8 establishes the claim for t=3. Suppose the claim is (n − 1) d true for some t \ 4. By application of Lemma B9, ct < 1 − 2n and dt < dn . Hence Lemma B5 applies, so the claim holds at time t+1. The result follows by induction. L 2n Lemma B11. Suppose that max{2(nn+1 − 1) , `(n − 1)(n − 2) } [ d [ 1. Then, ct Q 0 . and dt Q 0 as t Q . for the sequences {ct } t=3 and {dt } . t=3 constructed in Lemma B9.

Proof. Lemma B9 implies that {ct } . t=3 is monotonically decreasing on (n − 1) d [0, c3 ], where c3 =1 − 2n . Hence there must be a limit c g satisfying c g= g dc* g g max{dc* n , G(c , n ), H(c )}. Suppose for contradiction that c > 0. Since dc H(c) < c and n < c for all c ¥ [0, 1] the only possibility for c g > 0 is if dc 0 < c g=G(c g, dc* n ) [ c3 . After some algebra, the equation c=G(c, n ) can be rewritten as c

1 (n −2n1) d (d+3) − 1 2=(2n −n 3) d − 1,

(B26)

(2n − 3) d

where d \ 2(nn+1 − 1 > 0. It follows from (B26) that − 1) and n \ 5 S n (n − 1) d g (d+3) − 1 > 0 if c > 0 and the limit must be the (unique) fixed point 2n to (B26); i.e., (n − 32 ) (n − 1) d (n − 32 ) (2n − 3) d −1 4 −1 4 K−1 n n−1 2n n−1 g c = = = , (n − 1) d (n − 1) d K(d+3) − 1 (d+3) − 1 (d+3) − 1 2n 2n (B27) (n − 1) d

where K= 2n . In order for c g to be a possible limit to the sequence g {ct } . t=3 it must be that c [ c3 or

1 2

4(n − 32 ) 3 K−1 4 n− n−1 2 [ 1−K Z [ (d+3) − (K(d+3) − 1) K(d+3) − 1 n−1 Z K(d+3) − d [

2 . n−1

(B28)

349

LEGISLATIVE BARGAINING (n − 1) d 2n

Substituting back K= find that,

2n and using the condition that d \ `(n − 1)(n − 2) we

(n − 3) d 1 (n − 3) d (n − 1) 2n (n − 3) d (n − 1) d 2 + = + [ + 2n n−2 2n 2n(n − 1)(n − 2) 2n 2n =

(n − 1) d 2 (d+3) − d=K(d+3) − d [ 2n n−1

S

2 1 (n − 3) d [ − 2n n−1 n−2

=

(n − 3) 2(n − 2) − (n − 1) = (n − 1)(n − 2) (n − 1)(n − 2)

Sd[

2n < (n − 1)(n − 2)

2n =(n − 1)(n , − 2)

which establishes that any c g > 0 solving (B27) must be such that c g > c3 . Hence c g=0 is the only possible limit for {ct } . 3 . It follows immediately that limt Q . dt =dn limt Q . ct =0. L Proof of Proposition. Let x be such that x j > 0 for all j. First consider a history ht − 1 where x was proposed and rejected at time t. Then suppose that every i ¥ I follows the proposal rule

˛

p ijt− 1 (ht − 1 )=

1 − (n − 1) ct − 1 ct − 1

j=1 to all other agents.

(B29)

Since dt < ct for each t \ 3 and since ct Q 0 as t Q . Lemma B10 implies that (B29) is supportable as an equilibrium proposal independent of the identity of the proposer for t sufficiently large. If x is rejected, the discounted expected payoff for agent j=2, ..., n is dct − 1 Q 0 as t Q .. Hence there is some t large enough so that the voting rule ‘‘vote to accept x independently of the voting history’’ is sequentially rational. Hence, there are sequentially rational continuation strategies such that x is accepted if proposed in the first period. For histories where the proposer proposed y ] x, relabel the agents such that n is the deviant proposer at time t and y 1 [ y 2 [ · · · [ y n − 1. If y n=0, suppose that the next period proposal is in accordance with (B29). The proposer then either gets a payoff of zero (if the deviant proposal is accepted) or dct − 1 (if the deviant proposal is rejected), and for t large enough x n > dct − 1 , implying that no such deviation

350

PETER NORMAN

is profitable. If y n > 0 on the other hand, assume that every i ¥ I follows the proposal rule

˛

yi +ct − 1 d p ijt− 1 (ht − 1 )= y d max i +ct − 1 , d n ct − 1

3

to j=1, ...,

4

n−1 2

n+1 to j= 2 to all other agents

y

yj

(B30)

(n − 1) d

at time t − l. Suppose first that dj < dn. In this case ; (n+1)/2 and j=1 d < 2n since ct − 1 Q 0 as t Q . the proposal rule (B30) is feasible for large enough t. If yj /d \ dn , then we note thatn y n > 0 implies that ; (n+1)/2 y j < 2(nn+1 j=1 − 1) since 1−y 2 (n+1)/2 1 j j y [ n − 1 ;j y [ n − 1 . Again the proposal is feasible for t large n+1 ; j=1 enough and in both cases Lemma B10 implies that (B30) is supportable as an equilibrium proposal independent of the identity of the proposer for t sufficiently large. Moreover, each agent j [ n+1 receives a share p ij(ht ) > 2 j y /d independent of who is the proposer at time t. Hence, each j [ n+1 2 has a strict incentive to reject y, so the voting strategy ‘‘always vote to reject y’’ is sequentially rational for j=1, ..., n+1 2 , implying that for t large enough any y ] x is rejected in equilibrium. The expected discounted payoff for the proposer is dn ct − 1 if y is rejected and since dn ct − 1 Q 0 as t Q . there is some T < . such that there exists an equilibrium where each i is willing to propose x if recognized in the first period of the game with t \ T periods and where x is accepted by a majority. L

APPENDIX C: PROOF OF PROPOSITION 3 Proof of Proposition 3. In the last stage the unique equilibrium proposal is that the proposer takes everything for herself, so uniqueness holds trivially in the one-stage (ultimatum) game for all d. Now suppose that there is a unique equilibrium in the game with T − 1 periods, but that uniqueness fails when there are T periods. Uniqueness of equilibria in the game with T − 1 periods implies that there is a vector (V 1T − 1 , ..., V nT − 1 ) of continuation values such that V iT − 1 =V i(hT − 1 | s|T − 1 ) for every agent i and any subgame perfect equilibrium s (in the game with T − 1 or more rounds). But then (recall Fig. 2 in Section 5.1) a proposal where a majority gets strictly more than their respective discounted continuation values is accepted and any proposal where a majority gets strictly less is rejected if proposed at time T. For the usual reasons, acceptances are resolved in favor of acceptance in equilibrium, so if for each i the set I 0 {i} can be partitioned into two sets with n −2 1 agents each where any agent in one of the

351

LEGISLATIVE BARGAINING

sets has a strictly lower discounted continuation value than all agents in the other set, then the equilibrium is unique also in the game with T periods. Hence, a necessary condition for failure of uniqueness is that d iV iT − 1 = d jV jT − 1 for some i, j. If there is a unique equilibrium continuation V iT − 1 at time T − 1 for all i ¥ I there is a unique equilibrium continuation payoff V it for i ¥ I and each t < T − 1. If agents are labeled so that dV 1t < dV 2t < · · · < dV nt then (V 1t+1 , ..., V nt=1 ) can be expressed in terms (V 1t , ..., V nt ) according to

V

i t+1

˛

1 2 (n − 1) 1 = 1 1 − C d V 2+ dV n 2n 11 1− C d V 2 n n+1

2 1 n−1 i i 1+d iV it − C d jV jt = d Vt n n j=1 n−1 2

j

j t

j

j t

i

for i <

n+1 for i= 2

i t

j=1 n−1 2

n+1 2

for i >

j=1

(C1)

n+1 . 2

Now, pick any i ¥ I and treat d −i=(d 1, ..., d i − 1, d i+1, ..., d n) as a vector of constants. We can then express the continuation payoff at time t+1 in the form k V jt+1 =a jt +b jit d iV it + C b jk t Vt .

(C2)

k]i

Now, repeated substitution from (C2) gives k V jT =a jT − 1 +b jiT − 1 d iV iT − 1 + C b jk T−1VT−1 k]i

1

k =a jT − 1 +b jiT − 1 d i a iT − 2 +b iiT − 2 d iV iT − 2 + C b ik T−2VT−2

2

k]i

1

2

kj j i i + C b jk a kT − 2 +b ik T−1 T−2 d VT−2+ C bT−2VT−2 . k]i

(C3)

j]i

Collecting terms in (C3) we see that the value at time T can be written in k the form V jT =a˜ j+b˜ j1d i+b˜ j2(d i) 2 V iT − 2 +;k ] i b˜ jk T − 2 V T − 2 . Continuing the process recursively all the way back to t=1 we can eventually express the continuation value in the T period game for any agent j ¥ I as a polynomial of degree T − 1 in d i, V jT =a j+b j1d i+b j2(d i) 2+ · · · +b jT − 1(d i) T − 1.

(C4)

352

PETER NORMAN

I now claim that a j > 0 for all j. The reason for this is most easily understood from the procedure used to generate the polynomial. The coefficients in (C2) correspond with some ranking of the agents, say Pt , at time t. Hence (C4) can be generated by setting V j1 =1n for all j and picking a sequence of rankings PT − 1 , ..., P1 , and assume that in each period t [ T − 1 each agent bribes the n −2 1 lowest ranked agents according to the ranking Pt with their ‘‘continuation value’’ and keeps the rest. While this process is consistent with equilibrium only for particular rankings it generates a welldefined sequence of ‘‘values’’ that sum to unity in every period. But values summing to unity means that V jT > 0 for all j (by the use of (C1)). In particular, this means that the formula (C4) evaluated at d i=0 must be nonzero, which establishes that a j > 0. Next note that since the coefficients in (C2) depend on the rankings, they do depend on d i, but only to the extent that the ranking is changed. There are n! ways to order n agents and there are T − 1 recursions on (C2), implying that there is at most (n!) T − 1 sets of n polynomials to consider. 8 Now, if d iV iT =d jV jT , then we can use (C4) to write this as 0=d jV jT − d iV iT =d ja j+(d jb j1 − d ia i) d i+(d jb j2 − b i 1)(d i) 2+ · · · +(d jb jT − 1 − b iT − 2)(d i) T − 1 − b iT − 1(d i) T.

(C5)

Since (C5) has at least one coefficient different from zero there can be at most T roots to the polynomial. The set of polynomials to consider is bounded by (n!) T − 1 and there are n − 1 agents different from i, so given any ˜ d −i the number of values for d i that generate d iV iT =d jV jT for some j ] i is bounded by T(n − 1)(n!) T − 1, a finite number. Moreover, each coefficient of each polynomial is a sum of products of terms from the set {d 1, ..., d i − 1, −1 1 d i+1, ..., d n, 1, n2n , n}, so the coefficients are continuously differentiable in i −i d . Hence, X T ={d ¥ D | d iV iT =d jV jT for some j ¥ I 0 {i}} is a (n − 1)dimensional smooth manifold, a set with Lebesgue measure zero. Agent i was arbitrarily chosen, so we conclude that the set of discount factors such that some pair of discounted continuation values coincide, XT =1i ¥ I X iT , is also a negligible set. d ¥ X is a necessary condition for uniqueness to fail

8 While it doesn’t matter for the argument, observe that this is a gross overstatement of the number of polynomials to consider. In particular, it only matters whether an agent is the median, cheaper than the median, or more expensive than the median, so the number of relevant possibilities in each stage is no more than n((n − 1)!/( n −2 1 )! ( n −2 1 )!).

LEGISLATIVE BARGAINING

353

at time T so the critical values for which the T-period game has a nonunique equilibrium is a subset of 1 Tt=1 Xt , again a negligible set. We conclude that the set of d for which there is not a unique equilibrium has measure zero. L

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